In multiple dimensions, one simply performs this change of variables on each dimension separately, as desired, multiplying the integrand by the corresponding Jacobian factor for each dimension being transformed.

In multiple dimensions, one simply performs this change of variables on each dimension separately, as desired, multiplying the integrand by the corresponding Jacobian factor for each dimension being transformed.

-

The Jacobian factors diverge as the endpoints are approached. However, in order for your integral to have been finite in the first place it must be true that ''f''(''x'') goes to zero at infinity faster than 1/''x'', hence the limit of the integrand multiplied by the Jacobian is zero at the endpoints. In any case, the quadrature/cubature rules currently employed in <code>cubature.c</code> do not evaluate the integrand at the endpoints, so you need not implement special handling for |''t''|=1.

+

The Jacobian factors diverge as the endpoints are approached. However, if ''f''(''x'') goes to zero at least as fast as 1/''x''<sup>2</sup>, then the limit of the integrand (including the Jacobian factor) is finite at the endpoints. If your ''f''(''x'') vanishes more slowly than 1/''x''<sup>2</sup> but still faster than 1/''x'', then the integrand blows up at the endpoints but the integral is still finite (it is an integrable singularity), so the code will work (although it may take many function evaluations to converge). If your ''f''(''x'') vanishes only as 1/''x'', then it is not [[w:Absolute convergence|absolutely convergent]] and much more care is required even to define what you are trying to compute. (In any case, the quadrature/cubature rules currently employed in <code>cubature.c</code> do not evaluate the integrand at the endpoints, so you need not implement special handling for |''t''|=1.)

Revision as of 00:52, 14 May 2011

Contents

Cubature

Steven G. Johnson has written a simple C subroutine for adaptive multidimensional integration (cubature) of vector-valued integrands over hypercubes, i.e. to compute integrals of the form:

(Of course, it can handle scalar integrands as the special case where is a one-dimensional vector: the dimensionalities of and are independent.) The integrand can be evaluated for an array of points at once to enable easy parallelization. The code, which is distributed as free software under the terms of the GNU General Public License (v2 or later), is based on the algorithms described in:

This algorithm is best suited for a moderate number of dimensions (say, < 7), and is superseded for high-dimensional integrals by other methods (e.g. Monte Carlo variants or sparse grids).

(Note that we do not use any of the original DCUHRE code by Genz, which is not under a free/open-source license.) Our code is based in part on code borrowed from the HIntLib numeric-integration library by Rudolf Schürer and from code for Gauss-Kronrod quadrature (for 1d integrals) from the GNU Scientific Library, both of which are free software under the GNU GPL. (Another free-software multi-dimensional integration library, unrelated to our code here but also implementing the Genz–Malik algorithm among other techniques, is Cuba.)

I am also grateful to Dmitry Turbiner (dturbiner ατ alum.mit.edu), who implemented an initial prototype of the "vectorized" functionality (see below) for evaluating an array of points in a single call, which facilitates parallelization of the integrand evaluation.

Download

a gzippedtar file. This unpacks to a directory containing a README file with instructions and a stand-alone cubature.c file that you can compile and link into your program, and a header file cubature.h that you #include. (The cubature.c code is small enough that it doesn't seem worthwhile to go to the trouble of installing it as a library.)

The cubature.c file also contains a little test program which is produced if you compile that file with -DTEST_INTEGRATOR (and is commented out otherwise), as described below.

This integrates a function F(x), returning a vector of FDIM integrands, where x is a DIM-dimensional vector ranging from XMIN to XMAX (i.e. in a hypercube XMINi ≤ xi ≤ XMAXi).

MAXEVAL specifies a maximum number of function evaluations (0 for no limit). [Note: the actual number of evaluations may somewhat exceed MAXEVAL: MAXEVAL is rounded up to an integer number of subregion evaluations.] Otherwise, the integration stops when the estimated |error| is less than REQABSERROR (the absolute error requested) or when the
estimated |error| is less than REQRELERROR × |integral value| (the relative error requested). (Either of the error tolerances can be set to zero to ignore it.)

VAL and ERR are arrays of length FDIM, which upon return are the computed integral values and estimated errors, respectively. (The estimated errors are based on an embedded cubature rule of lower
order; for smooth functions, this estimate is usually conservative.)

The return value of adapt_integrate is 0 on success and nonzero if there was an error (currently, only out-of-memory situations).

Here, the input is an array X of length NDIM (the point to be evaluated), the output is an array FVAL of length FDIM (the vector of function values at the point X).

The FDATA argument of F is equal to the FDATA argument passed to adapt_integrate—this can be used by the caller to pass any additional information through to F as needed (rather than using global variables, which are not re-entrant). If F does not need any additional data, you can just pass FDATA = NULL and ignore the FDATA argument to F.

"Vectorized" interface

These integration algorithms actually evaluate the integrand in "batches" of several points at a time. It is often useful to have access to this information so that your integrand function is not called for one point at a time, but rather for a whole "vector" of many points at once. For example, you may want to evaluate the integrand in parallel at different points. This functionality is available by calling:

All of the arguments and the return value are identical to adapt_integrate, above, except that now the integrand F is of type integrand_v, corresponding to a function of a different form. The integrand function F should now be a function of the form:

Now, X is not a single point, but an array of NPTS points (length NPTS×NDIM), and upon return the values of all FDIM integrands at all NPTS points should be stored in FVAL (length FDIM×NPTS). In particular, x[i*ndim + j] is the j-th coordinate of the i-th point (i<npts and j<ndim), and the k-th function evaluation (k<fdim) for the i-th point is returned in fval[k*npt + i].

The size of NPTS will vary with the dimensionality of the problem; higher-dimensional problems will have (exponentially) larger NPTS, allowing for the possibility of more parallelism. Currently, NPTS starts at 15 in 1d, 17 in 2d, and 33 in 3d, but as adapt_integrate_v calls your integrand more and more times the value of NPTS will grow. e.g. if you end up requiring several thousand points in total, NPTS may grow to several hundred. We utilize an algorithm from:

Here, we have specified a relative error tolerance of 10 − 4 (and no absolute error tolerance or maximum number of function evaluations). Note also that, to demonstrate the fdata parameter, we have used it to pass the σ value through to our function (rather than hard-coding the value of σ in f or using a global variable).

The output should be:

Computed integral = 13.69609043 +/- 0.00136919

Note that the estimated relative error is 0.00136919/13.69609043 = 9.9969×10–5, within our requested tolerance of 10 − 4. The actual error in the integral value, as can be determined e.g. by running the integration with a much lower tolerance, is much smaller: the integral is too small by about 0.00002, for an actual relative error of about 1.4×10–6. As mentioned above, for smooth integrands the estimated error is almost always conservative (which means, unfortunately, that the integrator usually does more function evaluations than it needs to).

With the vectorized interface adapt_integrate_v, one would instead do:

Infinite intervals

Integrals over infinite or semi-infinite intervals is possible by a change of variables. This is best illustrated in one dimension.

To compute an integral over a semi-infinite interval, you can perform the change of variables x=a+t/(1-t):

For an infinite interval, you can perform the change of variables x=t/(1-t2):

Note the Jacobian factors multiplying f(⋅⋅⋅) in both integrals, and also that the limits of the t integrals are different in the two cases.

In multiple dimensions, one simply performs this change of variables on each dimension separately, as desired, multiplying the integrand by the corresponding Jacobian factor for each dimension being transformed.

The Jacobian factors diverge as the endpoints are approached. However, if f(x) goes to zero at least as fast as 1/x2, then the limit of the integrand (including the Jacobian factor) is finite at the endpoints. If your f(x) vanishes more slowly than 1/x2 but still faster than 1/x, then the integrand blows up at the endpoints but the integral is still finite (it is an integrable singularity), so the code will work (although it may take many function evaluations to converge). If your f(x) vanishes only as 1/x, then it is not absolutely convergent and much more care is required even to define what you are trying to compute. (In any case, the quadrature/cubature rules currently employed in cubature.c do not evaluate the integrand at the endpoints, so you need not implement special handling for |t|=1.)

where <dim> = # dimensions, <tol> = relative tolerance, <integrand> is 0–7 for one of eight possible test integrands (see below) and <maxeval> is the maximum number of function evaluations (0 for none, the default).

The different test integrands are:

0: a product of cosine functions

1: a Gaussian integral of exp(-x^2), remapped to [0,infinity) limits

2: volume of a hypersphere (integrating a discontinuous function!)

3: a simple polynomial (product of coordinates)

4: a Gaussian centered in the middle of the integration volume

5: a sum of two Gaussians

6: an example function by Tsuda, a product of terms with near poles

7: a test integrand by Morokoff and Caflisch, a simple product of dim-th roots of the coordinates (weakly singular at the boundary)

For example:

./cubature_test 3 1e-5 4

integrates the Gaussian function (4) to a desired relative error tolerance of 10–5 in 3 dimensions. The output is:

Notice that it finds the integral after 82203 function evaluations with an estimated error of about 10–5, but the true error (compared to the exact result) is much smaller (2.5×10–8): the error estimation is typically conservative when applied to smooth functions like this.